# SELF-DUALITY FOR LANDAU–GINZBURG MODELS BRIAN CALLANDER, ELIZABETH GASPARIM, ROLLO JENKINS AND

```JGSP 35 (2014) 1–10
SELF-DUALITY FOR LANDAU–GINZBURG MODELS
BRIAN CALLANDER, ELIZABETH GASPARIM, ROLLO JENKINS AND
LINO MARCOS SILVA
Communicated by Vasil V. Tsanov
Abstract. P. Clarke describes mirror symmetry as a duality between Landau–
Ginzburg models, so that the dual of an LG model is another LG model. We describe
examples in which the underlying space is a total space of a vector bundle on the
projective line, and we show that self-duality occurs in precisely two cases: the
cotangent bundle and the resolved conifold.
1. Introduction
For us a Landau–Ginzburg model (LG) is a variety X together with a regular function W : X → C called the superpotential. Clarke [1] showed that one can state
a generalised version of the Homological Mirror Symmetry conjecture of Kontsevich [4] as a duality between LG models. He also showed that this correspondence
generalises those of Batyrev–Borisov, Berglung–H&uuml;bsch, Givental, and Hori–Vafa.
This paper is an exercise in understanding the details of this correspondence. We
summarise the construction in [1], which, for a given LG model (X, W ), produces
a dual (X ∨ , W ∨ ). When (X ∨ , W ∨ ) ∼
= (X, W ), we call X self-dual. We then
study the case when X is the total space of a vector bundle on P 1 and prove that
self-duality occurs in only two cases: X = Tot(O(−2)) and X = Tot(O(−1) ⊕
O(−1)).
2. The Character to Divisor Map
Let X be a toric variety of rank n with a torus embedding ι : T −→ X. The torus
T = (C∗ )n is an algebraic group, whose algebraic functions are characters, that is,
group morphisms, χ : T −→ C∗ . Let M denote the group of characters of T , and
N the group of one-parameter subgroups, naturally identified with the dual of M ,
HomZ (M, Z). Let MR and NR denote the tensor products M ⊗Z R and N ⊗R Z,
respectively.
Since ι(T ) is dense inside X, each character χ ∈ M can be thought of as a rational
map, fχ : X 99K C, which is nowhere zero on ι(T ). Let R = {D1 , . . . , Dr }
doi: 10.7546/jgsp-35-2014-1-10
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